Download RANDOM NUMBERS, MONTE CARLO METHODS

RANDOM NUMBERS,
MONTE CARLO
METHODS
MA/CSC 428, SPRING 2017
NUMERICAL MATH II.
SAUER: CHAPTER 9.
TOO MANY APPLICATIONS TO LIST
1.
Simulation of natural, physical, and other complicated
phenomena (particles, people, traffic, …)
2.
Sampling from an impractically large number of possible
cases/scenarios to approximate “typical” or “average” behavior.
3.
Numerical integration can be seen as a special case of this.
4.
Computer science. For many (deterministic) algorithmic
problems, randomized algorithms provide the fastest (and
sometimes the only known efficient) solution. (Ex: testing
primes.)
5.
Secure communication, cryptography, privacy.
6.
Computer-generated imagery (CGI).
7.
Gambling and other recreation (duh).
Different applications utilize different properties of “random”
numbers.
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WHAT IS A RANDOM NUMBER?
Is 2.71947 a random number?
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WHAT IS A RANDOM NUMBER?
Is 2.71947 a random number? Not at all, but it was (seriously!)
randomly generated (by me).
It’s easier to meaningfully talk about a random set or sequence of
numbers following a particular probability distribution and about
random generation of numbers.
• The precise math details are hairy (421); we won’t need them.
Some intuitive interpretations of “random”:
• Uniformity: each number or item (from some domain or set) is
equally likely. More generally: the likelihood of different
outcomes is fixed, but not the outcomes (bell curve, etc.)
• Unpredictability: even looking at the initial numbers in a
sequence, we don’t know what comes next.
• Probability theory/statistics terminology: INDEPENDENT,
IDENTICALLY DISTRIBUTED (or IID) random variables.
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(PSEUDO-­)RANDOM
NUMBER SEQUENCES
What we usually need in numerical methods is
•
•
•
a sequence of independent, and
identically distributed
numbers from some domain, usually {0,1} or [0,1] or ℝ.
Computers rarely do truly random things, but for numerical math
applications we are usually happy with “random looking”,
deterministic, pseudo-random sequences.
• Uniformity (or following a prescribed distribution) is key
• True unpredictability is not so important, but at least apparent
statistical independence is. (This is very different from some
other applications, like cryptography, where strong
independence is essential!)
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(PSEUDO-­)RANDOM
NUMBER SEQUENCES
1. Uniformity is easy, independence is hard to achieve.
• (A quick puzzle: you have a biased coin. How can you generate
unbiased heads/tails with it?)
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(PSEUDO-­)RANDOM
NUMBER SEQUENCES
1. Uniformity is easy, independence is hard to achieve.
• (A quick puzzle: you have a biased coin. How can you generate
unbiased heads/tails with it?)
2. True uniform random is often not “as uniform as it could be”
1
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0
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1
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TWO SIMPLE RNGS
(Note: nobody uses these.)
The “middle-square” method (Neumann)
• Let’s say we’d like “random” 4-digit numbers.
1. Start with any 4-digit number (seed),
2. Square it, take the middle 4 digits as the next number,
3. Repeat.
• Example: seed=2017. 2017: = 04068289; 682: =
00465124, ; 4651: = 21631801, etc.
• 682, 4651, 6318, 9171, 1072, 1491, 2230, 9729, 6534, 6931, 387,
1497, 2410, 8081, 3025, 1506, 2680, 1824, 3269, 6863, …
• Looks random enough…(?)
• Their magnitudes are all over the place.
• 9 even and 11 odd numbers.
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TWO SIMPLE RNGS
(Note: nobody uses these.)
The linear congruential generator
•
•
•
•
Let’s say we’d like “random” integers from 0 to m-1.
Uses two parameters, integers a and b.
Start with any integer 𝑥] from 0 to m (seed).
𝑥P = 𝑎 𝑥P<J + 𝑏 (mod 𝑚)
• Example: seed=1, 𝑎 = 201, 𝑏 = 7, 𝑚 = 2017 (a prime)
• 208, 1475, 2000, 624, 377, 1155, 207, 1274, 1939, 465, 690,
1541, 1147, 616, 786, 667, 952, 1761, 993, 1934, …
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TWO SIMPLE RNGS
Some common features/bugs they share with more advanced
(pseudo-)random number generators:
• They are, of course, not random, completely deterministic,
• Require a seed (initialization),
• periodic, at least eventually.
• Can be easily modified to generate random bits or uniform
random (“real”) numbers from [0,1].
• The two most common applications, the next two being the
normal and exponential distributions, which are a bit more
complicated to do.
• Q: Will they be identically distributed? Will they be
independent?
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MATLAB
•
rng(seed) seeds the random number generator using the
nonnegative integer seed so that rand, randi, and randn
produce a predictable sequence of numbers.
•
•
•
rand: uniform real in [0,1].
randi(imax): uniform integer between 1 and imax.
randn: standard normal (mean 0, variance 1).
•
Alternative syntax available to generate arrays of random
numbers, including sparse arrays, random complex numbers,
random permutations, etc.
See the Matlab help (e.g., type “doc rand” for more details).
•
•
Reading and exercises: Sec. 9.1, including Exercises and
Computer Problems.
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MONTE CARLO SIMULATION
AND INTEGRATION
Simulation:
• Generate “representative” samples from a set of possible
scenarios.
• Simplest case: every scenario is equally likely.
• Simplest definition of representative: choose them with any
(pseudo-) RNG that covers the scenario space uniformly.
Application in integration:
• Remember this?
Q
K 𝑓 𝑥 𝑑𝑥 ≈ N 𝑤P 𝑓 𝑥P
c
PRJ
• Let 𝑥P be uniform random, and 𝑤P
=
J
∫ 1𝑑𝑥.
Q c
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MONTE CARLO INTEGRATION
J
Our previous example: compute ∫<J sin: 10𝜋 𝑥 + 1 𝑑𝑥 .
• Make sure everything in the code below makes sense:
f
n
xs
int
=
=
=
=
@(x)(sin(10*pi*x+1).^2);
100;
% number of samples
2*rand(n,1)-1;
% uniform in [-1,1]
sum(f(xs))*(2/n) % each weight is 2/n
• Using m=10, 100, 1000, …, 108 (after rng(2017) ), I got
int = 0.9151, 1.1658, 1.0258, 0.9972, 1.0007, 1.0004, 0.9999, 0.9998.
Questions:
• The error is, obviously, random. Can we quantify it?
• Why is this better than the quadrature formulas from 427?
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MONTE CARLO INTEGRATION
Why is this better than the quadrature formulas from 427?
1. Generalizes effortlessly to multivariate integration and (with
some work) to integration over complicated domains.
Q
K 𝑓 𝑥 𝑑𝑥 ≈ N 𝑤P 𝑓 𝑥P , 𝑤P =
c
PRJ
1
K 1 𝑑𝑥
𝑛 c
There’s nothing here about ℝ, except that we need to somehow
know the area (measure) of the domain X.
1.0
• E.g., what is the area of the shaded region,
defined by
J
d
0.5
0.0
J
≤ 𝑥 : + 𝑦 : ≤ 1 and ≤ 𝑥 : + 𝑦 : − 𝑥 𝑦 ≤ 1 ?
d
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0
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MONTE CARLO INTEGRATION
• What is the area of the shaded
region?
1
1
≤ 𝑥 : + 𝑦 : ≤ 1 and ≤ 𝑥 : + 𝑦 : − 𝑥 𝑦 ≤ 1
3
3
1.0
• Monte Carlo answer: generate a
number of random samples from
the enclosing square
• 𝑠ℎ𝑎𝑑𝑒𝑑 𝑎𝑟𝑒𝑎 = 0.5
# i7j PQ jk@lml # no 7n7@p inPQ7j
0.0
• Note that this is an RNG for
uniform distribution over the
shaded area.
• This is called the
acceptance-rejection method.
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0
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MONTE CARLO INTEGRATION
• Why random? Why not just use a grid?
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0.5
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MONTE CARLO INTEGRATION
• Why random? Why not just use a grid?
1.0
1. Random samples are
easier to scale and make
adaptive (e.g,. compare a 0.5
10x10 grid to a 11x11 grid).
2. With a full grid in higher
dimensions, we quickly
run out of points (≥2d)!
0.0
-0.5
3. Repeated experiments
provide statistical bounds
on the error.
-1.0
-1.0
-0.5
0.0
0.5
1.0
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QUICK RECAP I.
Monte Carlo integration
• The world’s simplest integration algorithm:
Q
K 𝑓 𝑥 𝑑𝑥 ≈ N 𝑤P 𝑓 𝑥P
c
PRJ
where
1
𝑤P = K 1𝑑𝑥
𝑛 c
and the 𝑥P are uniform random points from X.
• If X is complicated, we can generate these random points by
generating uniform random points from a simpler set containing
X (e.g., a box), and just drop the ones outside of X.
• With this method, as we are generating 𝑥P , we also get the
estimate
∫c 1𝑑𝑥
# 𝑝𝑡𝑠 𝑖𝑛 𝑋
≈
# 𝑝𝑜𝑖𝑛𝑡𝑠
∫=nr 1𝑑𝑥
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QUICK RECAP II.
Good, common (pseudo-)random number generators are
• Not random, but deterministic.
• The sequence is a function of the “seed” number. [rng()]
• Produce a periodic sequence of numbers (which is OK as
long as the period is really high).
• Are “apparently” random (pass simple statistical tests).
“Good” random number generation is quite difficult. (We
won’t be able to discuss the details.)
• For simple experiments, the Linear Congruential Generator
with a very long period is OK.
• But just use rand().
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MONTE CARLO INTEGRATION
• Why random? Why not just use a grid?
1. Random samples are
easier to scale and make
adaptive (e.g,. compare a
10x10 grid to a 11x11 grid).
2. With a full grid in higher
dimensions, we quickly
run out of points (≥2d)!
3. Repeated experiments
provide statistical bounds
on the error.
48
ERROR ESTIMATION
IN MONTE CARLO
Monte Carlo is an application of the law of large numbers.
• Informally, “the sample average converges to the mean.”
Monte Carlo is also an application of the central limit theorem.
• The average of iid samples approaches a normal distribution,
regardless of the distribution of the samples. (Under mild
assumptions.)
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ERROR ESTIMATION
IN MONTE CARLO
Monte Carlo is an application of the law of large numbers.
• Informally, “the sample average converges to the mean.”
Monte Carlo is also an application of the central limit theorem.
• The average of iid samples approaches a normal distribution.
The same is applicable to the errors:
• The sample average of the error converges to zero in a
“predictable” way:
•
𝑛
vw x⋯xvz
Q
average error
− 0 → 𝑁(0, 𝜎 : )
mean error
the variance of Ei (problem dependent!)
the “error of errors”
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ERROR ESTIMATION
IN MONTE CARLO
The order of magnitude of the error
• Rearranging 𝑛J/:
vw x⋯xvz
Q
− 0 = 𝑐𝑜𝑛𝑠𝑡,
• The error from an n-sample estimate of the integral,
is proportional to 𝑛<J/: .
vw x⋯xvz
,
Q
• Rearranging further: achieves an error <e using a number of
samples proportional to 1/𝜀 : .
• Q1: translate this to normal words?
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ERROR ESTIMATION
IN MONTE CARLO
The order of magnitude of the error
• Rearranging 𝑛J/:
vw x⋯xvz
Q
− 0 = 𝑐𝑜𝑛𝑠𝑡,
• The error from an n-sample estimate of the integral,
is proportional to 𝑛<J/: .
vw x⋯xvz
,
Q
• Rearranging further: achieves an error <e using a number of
samples proportional to 1/𝜀 : .
• Q1: translate this to normal words?
• Q2: is this a positive or negative result? J
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ERROR ESTIMATION
IN MONTE CARLO
The order of magnitude of the error
• Rearranging 𝑛J/:
vw x⋯xvz
Q
− 0 = 𝑐𝑜𝑛𝑠𝑡,
• The error from an n-sample estimate of the integral,
is proportional to 𝑛<J/: .
vw x⋯xvz
,
Q
• Rearranging further: achieves an error <e using a number of
samples proportional to 1/𝜀 : .
• Bad: Won’t yield tiny errors with few samples. Needs 100x
more samples for a single extra correct digit!
• Good: the error is dimension-independent!
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ERROR ESTIMATION
IN MONTE CARLO
The sample average only gives a single number (“point estimate”)
for the approximate integral.
• This means we don’t know how wrong or right the answer is.
A further application of the CLT and some statistics also yields
confidence intervals on the integral. (Try to do that with grids!)
• Main idea: group the samples into batches, calculate a set of
estimates, and based on their variability get a grip on the
(unknown) variance s2.
• Take a deep breath…
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ERROR ESTIMATION
IN MONTE CARLO
A further application of the CLT and some statistics also yields
confidence intervals on the integral.
• Main idea: group the samples into batches, calculate a set of
estimates, and based on their variability get a grip on the
(unknown) variance s2.
Theorem. If 𝑋J , … , 𝑋• ~ 𝑁(𝜇, 𝜎 : ) iid normal, then sample mean 𝑋„ =
cw x⋯xc… †8
~ 𝑁(𝜇, ) is also
•
•
J
∑•
deviation is 𝑆 =
•<J PRJ
normal. Moreover, if the sample standard
𝑋P − 𝑋„ : , then
𝑋„ − 𝜇
𝑇=
~ 𝑡•<J
𝑆/ 𝑚
Student’s t-­distribution with 𝑚 − 1 degrees of freedom
[The point is, this is a concrete, exactly known distribution, unlike 𝑁 0, 𝜎 : ].
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ERROR ESTIMATION
IN MONTE CARLO
Rearranging the final formula:
Corollary. A (1 − 𝑎)-confidence interval on µ is
𝑋„ ± 𝑡‹/:,•<J (𝑆/ 𝑚),
where 𝑡‹/:,•<J denotes the inverse CDF of the t distribution with
𝑚 − 1 degrees of freedom.
20
There is no nice formula for the t
values, but code and tables of important values are available.
Matlab: tinv(a/2,m-1)
15
10
5
0.2
0.4
https://onlinecourses.science.psu.edu/stat414/node/194
0.6
0.8
1.0
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ERROR ESTIMATION
IN MONTE CARLO
Example. (Phew…!)
1.0
• A confidence interval for p.
• MonteCarlo_pi.m
0.5
0.0
-0.5
-1.0
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-0.5
0.0
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QUASI-­MONTE CARLO (QMC)
Recall the pros and cons of (regular) grids:
•
•
•
It covers a rectangular domain nicely. (Really “uniform”.)
Can’t do adaptive/nested formulas easily.
Scales horribly with the dimension.
Recall the pros and cons of Monte Carlo:
•
•
•
Not really uniform, even if “uniform random”.
Can easily add any number of points.
Error is independent of the dimension.
Quasi-Monte Carlo: regular(ish) deterministic infinite sequence
whose initial finite subsequences cover the domain evenly.
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QUASI-­MONTE CARLO (QMC)
Quasi-Monte Carlo: regular(ish) deterministic infinite sequence
whose initial finite subsequences cover the domain evenly.
• Example in 1D (source:
http://planning.cs.uiuc.edu/node196.html):
• Quite regular and uniform
• Can you see the pattern?
• Called van der Corput sequence
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QMC: VAN DER CORPUT
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QMC: HALTON SEQUENCE
• The van der Corput sequence only works in 1D.
• If we use a (perhaps randomly shifted) vdC sequence in
each dimension, the coordinates are too obviously
correlated. (Not uniform.)
• Halton: use a different base for vdC along each coordinate.
7/8
1/16
8/9
5/9
2/9
4/9
1/9
2/3
1/3
7/9
T
8
3/8
T
6
5/8
T
7
1/8
T
4
3/4
T
5
1/4
T
3
1/2
T
1
1
x
T
2
0
0
y
1
1/16/17, 22:21
Source: wiki
61
MC VS QMC
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0
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0
0
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0.8
1
Which one is which?
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QMC: THE THEORY
The vdC and Halton sequences are examples of what is called a
low-discrepancy sequence.
•
Informally, the discrepancy of a point set measures how
uniformly the set covers the domain.
The discrepancy of an infinite sequence 𝑥J , 𝑥: , … measures
the asymptotic discrepancy of the finite subsequences
𝑥J , … , 𝑥Q as n grows.
•
•
Once again, the details are somewhat hairy. (See in a minute).
Short ref for the theory-inclined:
•
https://www.uibk.ac.at/mathematik/personal/einkemmer/qmm.pdf
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QMC: THE THEORY
Discrepancy of a point set 𝑥J , … , 𝑥Q in 0,1 l :
# inPQ7j r‘ PQ •
Q
•∈v
𝐷 (𝑥J , … , 𝑥Q ) = sup
− vol(𝐽) ,
where E is the set of rectangular boxes in 0,1 l .
• Intuitive measure of non-uniformity, but ultimately not so useful.
Instead, we have this:
Star-discrepancy (D*): the same as above, but E is the set of
rectangular boxes in 0,1 l with 0 as a vertex.
• Example: try and convince yourself that D* of the vdC sequence
–—˜ Q
is Θ(
).
Q
64